play store/google play the same or different?

Last night while looking to update apps, I clicked on the app "play store" that came with my tablet. None of the apps seemed to be listed there. I thought this. strange as I was sure they came from the Play Store. Finally, I googled for the apps and found them at Google.Play on the web!! And there my apps were and and all the buttons across the top that, for instance, had a button to show all of my apps!!! The Play Store app on my tablet had none of those. and the design was completely different. So what is up with that? can any one explain? I tried to see if the web Google,play had an app,but couldn't find it.

Same thing. If you open the Google Play Store app on your tablet, you'll notice three buttons top right.
Click the down arrow, it'll show you what you've downloaded ie what's on your machine.
The Magnifier next to it will search the Play Store.
The three wee buttons give you options/accounts.